One of the advantages of using the suggested prior in this paper is that the posterior is always twice continuously differentiable. In the estimation of sparse precision matrices, [1]} proposed the Bayesian version of graphical lasso, which uses Laplace approximation to compute posterior model probabilities of graphical structures and chooses the final model by MPM. [1]} considered Laplace prior on off-diagonal entries. Since Laplace prior is not differentiable at the median, Laplace approximation was applicable to only regular models. The term regularity may follow the terminology used by [3]} and [1]}. [3]} and [1]} put \(\ell _1\) -type penalty on the entries of the concentration matrix, which is the graphical lasso. When an off-diagonal entry is set as zero due to graphical lasso, the integrand in Laplace approximation becomes non-differentiable. If the model includes such an off-diagonal entry as a free variable, [3]} and [1]} referred to such model as a non-regular model, and a regular model otherwise. [1]} showed that the posterior probability of regular models is not smaller than that of their non-regular counterparts so that one can consider only regular models if one is to choose a model by MPM. However, one had to determine the regularity of each model and it turned out that the ratio of regular models among all possible models is extremely low. Moreover, the constraint regularity led the estimated matrix to be extremely sparse. On the contrary, since the posterior induced from the prior in this paper is always twice continuously differentiable, we do not have to consider the regularity. Thus, we do not have to suffer from the drawback of [1]} we just mentioned. Also, even though [1]} put a limit on the number of edges of a graph by Frequentist graphical lasso, there were still too many models to search over and one had to judge the regularity of each model which yields inefficiency in choosing the model. However, by using the algorithm in [12]}, we do not have to search over all models. Furthermore, since we force zeros on off-diagonal entries, sparsity can be naturally introduced to the structure of covariance matrix compared to those with continuous spike and slab prior in [13]} and [14]}. Hence, our prior is more useful in introducing sparsity to the covariance structure.
